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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sibfima | Structured version Visualization version GIF version |
Description: Any preimage of a singleton by a simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.) |
Ref | Expression |
---|---|
sitgval.b | β’ π΅ = (Baseβπ) |
sitgval.j | β’ π½ = (TopOpenβπ) |
sitgval.s | β’ π = (sigaGenβπ½) |
sitgval.0 | β’ 0 = (0gβπ) |
sitgval.x | β’ Β· = ( Β·π βπ) |
sitgval.h | β’ π» = (βHomβ(Scalarβπ)) |
sitgval.1 | β’ (π β π β π) |
sitgval.2 | β’ (π β π β βͺ ran measures) |
sibfmbl.1 | β’ (π β πΉ β dom (πsitgπ)) |
Ref | Expression |
---|---|
sibfima | β’ ((π β§ π΄ β (ran πΉ β { 0 })) β (πβ(β‘πΉ β {π΄})) β (0[,)+β)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sibfmbl.1 | . . . 4 β’ (π β πΉ β dom (πsitgπ)) | |
2 | sitgval.b | . . . . 5 β’ π΅ = (Baseβπ) | |
3 | sitgval.j | . . . . 5 β’ π½ = (TopOpenβπ) | |
4 | sitgval.s | . . . . 5 β’ π = (sigaGenβπ½) | |
5 | sitgval.0 | . . . . 5 β’ 0 = (0gβπ) | |
6 | sitgval.x | . . . . 5 β’ Β· = ( Β·π βπ) | |
7 | sitgval.h | . . . . 5 β’ π» = (βHomβ(Scalarβπ)) | |
8 | sitgval.1 | . . . . 5 β’ (π β π β π) | |
9 | sitgval.2 | . . . . 5 β’ (π β π β βͺ ran measures) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | issibf 33631 | . . . 4 β’ (π β (πΉ β dom (πsitgπ) β (πΉ β (dom πMblFnMπ) β§ ran πΉ β Fin β§ βπ₯ β (ran πΉ β { 0 })(πβ(β‘πΉ β {π₯})) β (0[,)+β)))) |
11 | 1, 10 | mpbid 231 | . . 3 β’ (π β (πΉ β (dom πMblFnMπ) β§ ran πΉ β Fin β§ βπ₯ β (ran πΉ β { 0 })(πβ(β‘πΉ β {π₯})) β (0[,)+β))) |
12 | 11 | simp3d 1143 | . 2 β’ (π β βπ₯ β (ran πΉ β { 0 })(πβ(β‘πΉ β {π₯})) β (0[,)+β)) |
13 | sneq 4638 | . . . . . 6 β’ (π₯ = π΄ β {π₯} = {π΄}) | |
14 | 13 | imaeq2d 6059 | . . . . 5 β’ (π₯ = π΄ β (β‘πΉ β {π₯}) = (β‘πΉ β {π΄})) |
15 | 14 | fveq2d 6895 | . . . 4 β’ (π₯ = π΄ β (πβ(β‘πΉ β {π₯})) = (πβ(β‘πΉ β {π΄}))) |
16 | 15 | eleq1d 2817 | . . 3 β’ (π₯ = π΄ β ((πβ(β‘πΉ β {π₯})) β (0[,)+β) β (πβ(β‘πΉ β {π΄})) β (0[,)+β))) |
17 | 16 | rspcv 3608 | . 2 β’ (π΄ β (ran πΉ β { 0 }) β (βπ₯ β (ran πΉ β { 0 })(πβ(β‘πΉ β {π₯})) β (0[,)+β) β (πβ(β‘πΉ β {π΄})) β (0[,)+β))) |
18 | 12, 17 | mpan9 506 | 1 β’ ((π β§ π΄ β (ran πΉ β { 0 })) β (πβ(β‘πΉ β {π΄})) β (0[,)+β)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 βwral 3060 β cdif 3945 {csn 4628 βͺ cuni 4908 β‘ccnv 5675 dom cdm 5676 ran crn 5677 β cima 5679 βcfv 6543 (class class class)co 7412 Fincfn 8942 0cc0 11113 +βcpnf 11250 [,)cico 13331 Basecbs 17149 Scalarcsca 17205 Β·π cvsca 17206 TopOpenctopn 17372 0gc0g 17390 βHomcrrh 33272 sigaGencsigagen 33435 measurescmeas 33492 MblFnMcmbfm 33546 sitgcsitg 33627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-sitg 33628 |
This theorem is referenced by: sibfinima 33637 sitgfval 33639 sitgclg 33640 |
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